A new type of boundary value coupling for second order Sturm-Liouville systems

A natural genera lization of the familiar second order Sturm-Liouville syste m is presented. T his gene rali za tion co ns is ts of considering a number of differe ntia l equ ations defined on difTerent inte rvals, instead of ju st one equation on one inte rva l. The self-adjoint characte r of the diffe rential equations is re ta ined in th e gene rali za tion, but th e boundary conditions a re re laxed conside ra bl y. The mos t gene ral boundary cond itions which can be accommod ated by thi s so rt of gene ralization of S turm-Liouvi ll e theory a re di scussed. Th e exis te nce of e igenva lues is proved , and a gene ra lized orthogonalit y and a weak e igenfun ction expansion theorem are de rived.